Mathematics Tutoring Resources

This resource is a compilation of good-quality videos that explain a variety of common topics. Please let us know if you find any links that aren't working.

Please note that we used two main Youtube channels (mathbff and KristaKingMath) that offer high quality information. We've noted the name of the channel along with the linked topic. You can certainly navigate to other videos on those channels right on Youtube, or you can use this page as a convenient way to identify just what you're looking for. By using this page, though, you help us determine how useful our providing this information is.

IMPORTANT NOTE: Professor Leonard states that the domain of a combination of functions is the intersection of the domains of the originals. While mostly true, there is an additional restriction when a combination of functions results in a fraction. In addition to finding the intersection of the original domains, you must also look for anywhere that your new function would have a zero in its denominator. Any such points must be excluded from the overall domain.

NOTE The beginning of this video provides a “conversational” address to the purpose of calculus. Examples are given, but technicalities are initially skipped over to provide more of a general idea rather than a more mathematical understanding. To go straight to the more technical, mathematical approaches and explanations, skip ahead to 39:21.

57:30 One-sided limits and determining whether or not a limit exists at a point

NOTE: Starting at 1:25:13, Professor Leonard answers a question from a student. The question is whether or not a limit technically exists if the limits from the left and from the right are either both positive or both negative infinity. His answer is:

If the limits from the left and from the right are both positive infinity, then the limit DOES exist, and it is positive infinity.

If the limits from the left and from the right are both negative infinity, then the limit DOES exist, and it is negative infinity.

In situations such as this, the limits from the left and from the right both approach the same thing, but it is important to remember that infinity is not a number. Infinity is merely a concept, and it is the idea of values that get larger and larger without end. Since infinity by itself is not a number, the situations described above cannot be said to have one-sided limits that approach the same value. That being said, while you CAN say that the limit is positive (or negative) infinity, you CANNOT say that the limit actually exists. In a situation such as limit, the limit is DNE (Does Not Exist), and you are said to have an infinite discontinuity.

NOTE: at 17:58, Professor Leonard says that the number you are evaluating a limit at will always be given. This is true, but his use of the word “number” at that point may be misleading. In the example at that point in the video, he has evaluated a limit for when approaches 2, but you can use the same methods to evaluate a limit if is approaching , or if x is approaching , or if x is approaching . To generalize what Professor Leonard says there, you will always be given what the variable in your limit problem is approaching, but be aware that what you are given will not always be a number.

(NOTE: This section of the video explains the identity . The explanation is complicated and you will generally not be asked to replicate it, but it is used to introduce and illustrate the Squeeze Theorem, so it worth the time to work through and understand.)

(NOTE: This section of the video explains the identity . As previously, you generally will not be asked to replicate the explanation, but it is still good practice to follow along with what Professor Leonard does so you can see some of the “tricks” that he uses to evaluate limits.)

(NOTE: This section of the video explains the identity . As with the last section, this section illustrates methods of algebraic manipulation that can be used when evaluating limits, so please follow along and notice what Professor Leonard does. You MIGHT be asked to replicate or recreated this proof, so make sure you understand how he arrives at his answer.)

2:06:29 The remainder of this video consists of examples of algebraic manipulation, the previous three identities, and the Squeeze Theorem being used to evaluate limits. These are methods and manipulations that you WILL be asked to apply on your own. It is VERY important that you follow along with and try to understand each problem that he does. After watching him do each problem, try pausing the video and recreating it on your own to test your recall and understanding.